Updated with data available as of 2020-05-20
Demonstrating model fit against COVID-19 data for Los Angeles, for the following variables/compartments:
COVID-19 data is shown as black dots in the figures below.


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Projections for key compartments:





















What a stochastic model allows:
Compartmental model flow diagram
\[ \begin{align*} dS/dt &= -\beta S(I+A)\\ dE/dt &= \beta S(I+A) - \tfrac{1}{d_{EI}}E\\ dA/dt &= \tfrac{1-r}{d_{EI}}E - \tfrac{1}{d_{IR}}A\\ dI/dt &= \tfrac{r}{d_{EI}}E - (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I\\ dH/dt &= \alpha (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I - (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H \\ dQ/dt &= \kappa (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H - (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q \\ dV/dt &= p_V Q\\ dD/dt &= \delta (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q\\ dR/dt &= (1-\alpha) (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I + (1-\kappa) (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H + (1-\delta)(\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q + \tfrac{1}{d_{IR}}A \ \end{align*} \]
\[ R0 = \beta ({\frac{r}{\tfrac{\alpha}{d_{IH}}+\tfrac{1-\alpha}{d_{IR}}}+ (1-r){d_{IR}}}) \\ N=S+E+A+I+H+Q+D+R \]
| Parameter | Description | Value |
|---|---|---|
| \(R0\) | Basic reproductive number | Estimated |
| \(\beta\) | transmission rate | Analytically derived from model and R0 |
| \(d_{EI}\) | days between exposure and infectivity (incubation period) | 5 days |
| \(d_{IH}\) | days between symptom onset and hospitalization (if required) | 10 days |
| \(d_{IR}\) | days between symptom onset and recovery (if not hospitalized) | 7 days |
| \(d_{HQ}\) | days between hospitalization and ICU (if required) | 1 days |
| \(d_{QR}\) | days between hospitalization and recovery (if ICU not required) | 12 days |
| \(d_{QD}\) | days between ICU and fatality | 8 days |
| \(d_{QR}\) | days between ICU and recovery | 7 days |
| \(\alpha\) | probability infected (I) requires hospitalization (vs. recovers) | Estimated |
| \(\kappa\) | probability hospitalized (H) requires ICU (vs. recovers) | Estimated |
| \(\delta\) | probability ICU (Q) patient dies | Estimated |
| \(p_V\) | probability ventilation (V) required given ICU | Estimated |
| \(N\) | Total population size | |
| \(S\) | Susceptible population | |
| \(E\) | Exposed not yet infectious | |
| \(A\) | Infected, unobserved | |
| \(I\) | Infected, observed | |
| \(H\) | In Hospital | |
| \(Q\) | In ICU | |
| \(V\) | On ventilator | |
| \(D\) | Dead | |
| \(R\) | Recovered/removed |
Model parameters - fixed, taken from literature: - Transition times between compartments - Sources provided at this link
Model Parameters — estimated by our model - \(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population - \(r\), the proportion of illnesses that are observed - \(Frac_{R0}\), the reduction in the initial R0 due to social distancing - \(\alpha\), the probability of hospitalization given illness, i.e. \(Pr(Hospital | Illness)\) - \(\kappa\), the probability of ICU care necessary given hospitalization, i.e. \(Pr(ICU | Hospital)\) - \(p_v\), the probability of ventilation given ICU care, i.e. \(Pr(Ventilation | ICU)\) - \(\delta\), the probability of death given ICU care, i.e. \(Pr(Death | ICU)\)
Model estimated parameters: Prior information and posterior estimates Here we summarize our estimated parameter values for key epidemic and model quantities:
Because our model is stochastic and we are using Bayesian techniques for parameter estimation, each posterior parameter estimate is represented by a distribution of likely values.
Summary of key statistics of each estimated parameter: the mean and the standard deviation (sd).
| R0 | Prop. cases detected (r) | Frac R0 Mar11 | Pr(Death|ICU) | Pr(Hospital|Illness) | Pr(ICU|Hospital) | Pr(Ventilation|ICU) | |
|---|---|---|---|---|---|---|---|
| mean | 4.1649 | 0.0267 | 0.2309 | 0.8537 | 0.2602 | 0.3012 | 0.5117 |
| sd | 0.2487 | 0.0046 | 0.0539 | 0.1214 | 0.0458 | 0.0412 | 0.0541 |
We use previous studies to narrow the specification of the probability of hospitalization given illness, admittance to the intensive care unit (ICU) given being in hospital, ventilation given being in ICU, and death given being in ICU by incorporating risk factors, including age, sex, smoking and other comorbidities. The prevalence of these risk factors in Los Angeles County is also included.
The comorbidities included in the “any comorbidity” category are: hypertension, diabetes, cardiovascular disease, cerebrovascular disease / stroke, cancer, COPD, and athsma. It is important to note that each of these conditions has its own risk of disease outcome; future models will disaggregate risk estimates by specific risk conditions when sufficient data becomes available to produce these estimations.
(1) Estimating the conditional probability of COVID illness severity given combinations of risk factors
\[ \begin{align*} Pr(Hospital | Illness) = \sum_i Pr(Group_i | Illness)Pr(Hospital|Group_i,Illness) \end{align*} \] We assume that the prevalence of the risk group in the ill population, \(Pr(Group_i|Illness)\), is equal to the prevalence of the group in the general population of L.A. County, i.e. \(Pr(Group_i)\). We again borrow the correlation structure between risk factors derived from the NHANES cohort to estimate the population prevalence \(Pr(Group_i)\) from available data on the prevalence of individual risk factors.
The same approach is applied to estimate \(Pr(ICU | Hospital)\) and \(Pr(Death|ICU)\): \[ \begin{align*} Pr(ICU | Hosptial) = \sum_i Pr(Group_i | Hospital)Pr(ICU|Group_i,Hospital)\\ Pr(Death | ICU) = \sum_i Pr(Group_i | ICU)Pr(Death|Group_i,ICU) \end{align*} \]
(2) Estimating the proportion of each risk group that will make up the cohorts of COVID patients admitted to hospital, admitted to ICU, or that die in L.A. County and SPAs
Data sources
COVID-19 Illness Trajectory Relative Risks
Studies on COVID-19 clinical presentation and trajectories to inform the probability of hospitalization, ICU, and ventilation based on single risk factors:
L.A. County Risk Factor Prevalences - Los Angeles County Health Survey - UCLA California Health Information Survey
Standard deviation = 0.249
\(R0\) prior estimate is based on values for \(R0\) estimated from other published studies on COVID-19.
Prior and posterior 
Standard deviation = 0.005
We use this information to inform two pieces of information: (i) the prior distribution for \(r\), the fraction of observed illnesses over all illnesses, and (ii) an observed data point for the number of counts in the Recovered compartment. We make the following assumptions:
Prior distribution for \(r\) - The seroprevalence study retuns an estimated % of the population of L.A. that has antibodies for COVID-19 by the study date. We assume this is equal to the proportion of total recovered individuals as of a week prior to the study date, April 4, 2020, since it takes a week for antibodies to develop. - Meanwhile, we assume the fraction of observed illnesses \(r\) is approximately equal to the fraction of recovered cases as:
\[ \begin{align*} \mathbf{r} = \frac{cum. obs. I(t = t')}{cum. tot. I(t=t')}\\ \approx \frac{cum. obs. R(t=t')}{cum.tot.R(t=t')}\\ \end{align*} \] - The seroprevalence study informs the denominator, the total recovered cases at \(t'\) = April 4. - The numerator, the “observed” recovered cases at time \(t'\), is assumed to be approximately equal to the observed illnesses at time \(t'-\) [1,3] weeks, which is the assumed period from infectiousness to recovery. - We specify the minimum and maximum values for the prior for \(r\) \(Pr(r)\) using the equation above with the range of the % of the population with antibodies on of April 4, and the number of observed illnesses 1-3 weeks prior to that. - In addition to using the seroprevalence study to inform the prior distribution on \(r\), we use it to inform the
Data point on number in Recovered compartment
We use the results of the seroprevalence study to inform a fixed data point representing the number of counts in the Recovered compartment on April 4. We use this data point together with the approximated timing of the first recovered case (1-3 weeks after the start date of the outbreak) to interpolate an exponential distribution in the time series of counts in the Recovered compartment. This is then used as an observed data input alongside the other observed compartment variables (illnesses, hospitalizations, deaths, etc.) in the parameter estimation procedure.
These estimates will be greatly improved by time series of COVID-19 seroprevalence, i.e. as results of future seroprevalence studies become available.
Prior and posterior 

We extend the risk factor analysis above to estimate the proportion of each risk group that will make up the cohorts of COVID patients admitted to hospital, admitted to ICU, or that die in L.A. County, across the SPAs, and across the race/ethnicity groups
Estimating the conditional probability of COVID illness severity given combinations of risk factors

(2) Social distancing scenarios
SCENARIOS EVALUATED: Increase contact rate on May 16th, 2020 by variable amounts
Scenarios evaluated
7 are illustrated here. Each scenario corresponds to a different amount of contact rate, and can be understood in terms of:
Maintain current level of social distancing
Equal to:
% of original R0: 35.8
Equal to:
% of original R0: 48.7
Equal to:
% of original R0: 61.5
Equal to:
% of original R0: 74.3
Equal to:
% of original R0: 87.2
Equal to:
% of original R0: 100
Equal to: